Oriented cohomology and invariants of homogeneous spaces

线性代数群理论是现代数学中一个久负盛名的领域。它最初是李群理论的一个代数版本，该理论获得了巨大的成功和广泛的应用，最著名的是由切瓦利和博雷尔推动的。在Serre、Springer、Tits等人的手中，它发展成为理解几何(齐次空间和环面簇的)、数论(伽罗华上同调的形式)和表示论(群及其相关代数)的重要工具。在过去的几十年里，线性代数群理论见证了代数拓扑学方法的大规模入侵。这些新方法在代数中的一些经典问题上取得了突破，这些问题是早期纯粹的代数技术所无法企及的。90年代中期，S·沃沃茨基将同伦和协边理论中的技巧运用到二次型中，解决了米尔纳猜想。这一趋势的另一个显著例子是Eddidin-Graham和Totaro发明了代数分类空间和等变上同调。将这些结果与Levine-Morel和Panin-Smirnov在00‘S中提出的面向代数上同调的概念相结合，产生了一类新的代数等变上同调理论(如等变上同调和椭圆上同调)，鉴于其与几何的丰富联系，这些理论目前得到了活跃的研究。粗略地说，它由两个方向组成：第一，是(在代数闭域上)等变定向上同调的‘代数化程序’；第二，是它在(任意域上)扭矩和扭曲旗簇理论中的应用。它的目标是将旗簇的各种上同调环和它们的经典感兴趣的元素(如舒伯特簇的类)与一些代数和组合对象相匹配。一些子任务，例如那些与有向上同调中的计算有关的任务，有望成为拟议的硕士和博士项目的主题。*作为结果，我们希望在代数群论和齐次空间几何方面获得新的结果，这将扩展我们在这些数学领域的知识，并将有助于提高加拿大的基础科学能力。

The theory of linear algebraic groups is a well established area of modern mathematics. It started as an algebraic version of the massively successful and widely applied theory of Lie groups, pushed forward most notably by Chevalley and Borel. In the hands of Serre, Springer, Tits and many others, it developed into an important tool for understanding geometry (of homogeneous spaces and toric varieties), number theory (in the form of Galois cohomology) and representation theory (of groups and the associated algebras). In the last decades the theory of linear algebraic groups has witnessed a massive intrusion of the methods of algebraic topology. These new methods have led to breakthroughs on a number of classical problems in algebra, which are beyond the reach of earlier purely algebraic techniques. In the mid of 90's Voevodsky's use of techniques from homotopy and cobordism theory in the context of quadratic forms had resulted in the solution of the Milnor conjecture. Another striking example of this ongoing trend is the invention of an algebraic classifying space and equivariant cohomology by Eddidin-Graham and Totaro. Merging these results with a notion of an algebraic oriented cohomology introduced by Levine-Morel and Panin-Smirnov in 00's have lead to the creation of a family of new algebraic equivariant cohomology theories (e.g. equivariant cobordism and elliptic cohomology) which are actively studied nowadays in view of its rich connections to geometry.******The proposed project can be viewed as the next step toward this philosophy. Roughly speaking, it consists of two directions: first, is an 'algebraization program' for equivariant oriented cohomology (over an algebraically closed field); second, deals with its applications to the theory of torsors and twisted flag varieties (over an arbitrary field). Its goal is to match various cohomology rings of flag varieties and elements of classical interest in them (such as classes of Schubert varieties) with some algebraic and combinatorial objects. An essential part of the proposed investigations is the involvement of graduate and postdoctoral students. Some of the subtasks, for instance those related with computations in oriented cohomology, are expected to become the subjects of the proposed MSc and PhD-projects.***As an outcome we expect to obtain new results in the theory of algebraic groups and geometry of homogeneous spaces that will extend our knowledge in those areas of mathematics and will help advance Canada's fundamental science capabilities. **